Towards Unavoidable Minors of Binary 4-connected Matroids

<p>We show that for every n ≥ 3 there is some number m such that every 4-connected binary matroid with an M (K3,m)-minor or an M* (K3,m)-minor and no rank-n minor isomorphic to M* (K3,n) blocked in a path-like way, has a minor isomorphic to one of the following: M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, or a rank-n matroid closely related to the cycle matroid of a double wheel, which we call a non graphic double wheel. We also show that for all n there exists m such that the following holds. If M is a 4-connected binary matroid with a sufficiently large spanning restriction that has a certain structure of order m that generalises a swirl-like flower, then M has one of the following as a minor: a rank-n spike, M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, a rank-n non graphic double wheel, M* (K3,n) blocked in a path-like way or a highly structured 3-connected matroid of rank n that we call a clam.</p>

Towards Unavoidable Minors of Binary 4-connected Matroids

<p>We show that for every n ≥ 3 there is some number m such that every 4-connected binary matroid with an M (K3,m)-minor or an M* (K3,m)-minor and no rank-n minor isomorphic to M* (K3,n) blocked in a path-like way, has a minor isomorphic to one of the following: M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, or a rank-n matroid closely related to the cycle matroid of a double wheel, which we call a non graphic double wheel. We also show that for all n there exists m such that the following holds. If M is a 4-connected binary matroid with a sufficiently large spanning restriction that has a certain structure of order m that generalises a swirl-like flower, then M has one of the following as a minor: a rank-n spike, M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, a rank-n non graphic double wheel, M* (K3,n) blocked in a path-like way or a highly structured 3-connected matroid of rank n that we call a clam.</p>

The connectivity and Hamiltonian properties of second-order circuit graphs of wheel cycle matroids

Let [Formula: see text] be the [Formula: see text]th-order circuit graph of a simple connected matroid M. The first-order circuit graph is also called a circuit graph. There are lots of results about connectivity and Hamiltonian properties of circuit graph of matroid, while there are few related results on the second-order circuit graph of a matroid. This paper mainly focuses on the connectivity and Hamiltonian properties of the second-order circuit graphs of the cycle matroid of wheels. It determines the minimum degree and connectivity of these graphs, and proves that the second-order circuit graph of the cycle matroid of a wheel is uniformly Hamiltonian.

Obstructions for Bounded Branch-depth in Matroids

DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid Un;2n or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.

A Splitter Theorem for 3-Connected 2-Polymatroids

Seymour's Splitter Theorem is a basic inductive tool for dealing with $3$-connected matroids. This paper proves a generalization of that theorem for the class of $2$-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. A series compression in such a structure is an analogue of contracting an edge of a graph that is in a series pair. A $2$-polymatroid $N$ is an s-minor of a $2$-polymatroid $M$ if $N$ can be obtained from $M$ by a sequence of contractions, series compressions, and dual-contractions, where the last are modified deletions. The main result proves that if $M$ and $N$ are $3$-connected $2$-polymatroids such that $N$ is an s-minor of $M$, then $M$ has a $3$-connected s-minor $M'$ that has an s-minor isomorphic to $N$ and has $|E(M)| - 1$ elements unless $M$ is a whirl or the cycle matroid of a wheel. In the exceptional case, such an $M'$ can be found with $|E(M)| - 2$ elements.

The magnitude of a graph

The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its cardinality-like properties are multiplicativity with respect to cartesian product and an inclusion-exclusion formula for the magnitude of a union. Formally, the magnitude of a graph is both a rational function over ℚ and a power series over ℤ. It shares features with one of the most important of all graph invariants, the Tutte polynomial; for instance, magnitude is invariant under Whitney twists when the points of identification are adjacent. Nevertheless, the magnitude of a graph is not determined by its Tutte polynomial, nor even by its cycle matroid, and it therefore carries information that they do not.

A General Method for Kinematic Analysis of Robotic Wrist Mechanisms

The paper presents a novel and simple technique for the kinematic analysis of bevel gear trains (BGT). The approach is based on edge-oriented graphs for efficient computation of BGT’s absolute and relative velocities of links using incidence matrices. The kinematic equations are generated in matrix form using a cycle basis from a cycle matroid. The set of independent equations is automatically obtained from matrix orthogonalities and not by taking derivatives. Equation coefficients are expressed as function of speed ratios and have minimal variables. Then the relationships between the output and input angular velocities can be determined. In addition, a simple procedure is demonstrated to check for mechanism singularities. The method presented here has general applicability and can be employed for spatial geared mechanisms with any number of gears and degrees of freedom (DOF) as illustrated by numerical examples of robotic wrist mechanisms.

Matroid and Tutte-Connectivity in Infinite Graphs

We relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.

Covering Cycle Matroid

Covering is a type of widespread data representation while covering-based rough sets provide an efficient and systematic theory to deal with this type of data. Matroids are based on linear algebra and graph theory and have a variety of applications in many fields. In this paper, we construct two types of covering cycle matroids by a covering and then study the graphical representations of these two types of matriods. First, through defining a cycle graph by a set, the type-1 covering cycle matroid is constructed by a covering. By a dual graph of the cycle graph, the covering can also induce the type-2 covering cycle matroid. Second, some characteristics of these two types of matroids are formulated by a covering, such as independent sets, bases, circuits, and support sets. Third, a coarse covering of a covering is defined to study the graphical representation of the type-1 covering cycle matroid. We prove that the type-1 covering cycle matroid is graphic while the type-2 covering cycle matroid is not always a graphic matroid. Finally, relationships between these two types of matroids and the function matroid are studied. In a word, borrowing from matroids, this work presents an interesting view, graph, to investigate covering-based rough sets.

Kinematic Analysis of Epicyclic Bevel Gear Trains With Matroid Method

The paper presents a novel technique for the kinematic analysis of bevel gear trains using the incidence matrices of an edge-oriented graph of the mechanism. The kinematic equations are then obtained in matrix form using a cycle basis from a cycle matroid. These equations can be systematically generated, and allow for an efficient computation of the angular velocities of the gears and planet carriers of the mechanism without employing time derivative operations. As illustrated in the paper, the method is applicable to bevel gear trains of any number of gears or degrees of freedom.